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Chapter 9

The Capital Asset Pricing Model. Chapter 9. Capital Asset Pricing Model (CAPM). It is the equilibrium model that underlies all modern financial theory. Derived using principles of diversification with simplified assumptions.

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Chapter 9

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  1. The Capital Asset Pricing Model Chapter 9

  2. Capital Asset Pricing Model (CAPM) • It is the equilibrium model that underlies all modern financial theory. • Derived using principles of diversification with simplified assumptions. • Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.

  3. Assumptions • Individual investors cannot affect prices by their individual trades. • Single-period investment horizon. • Investors form portfolios from many publicly traded financial assets. • They can borrow at Rf and have unlimited borrowing and lending opportunities • No taxes and transaction costs.

  4. Assumptions (cont’d) • Information is costless and available to all investors. • Investors are rational mean-variance optimizers. They attempt to construct Markowitz’s efficient frontier. • There are homogeneous expectations. All investors analyse the securities in the same way and share the same economic view of the world. All investors use the same expected return, standard deviations, and correlations to generate the efficient frontier and the unique optimal risky portfolio.

  5. Resulting Equilibrium Conditions • The proportion of the each security in all investors porfolios will be same as the proportion of each security in the market porfolio (consists of all securities in the market). All investors will hold the same portfolio for risky assets – market portfolio. Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value.

  6. Resulting Equilibrium Conditions 2. The market portfolio will be on the efficient frontier. It will be the optimal port, the tangency point of the CAL to the efficient frontier. As a result CML, the line from the rf through the market port, M is also the best attainable CAL. All investors hold M as the optimal risky portfolio, differing only in the amount invested in it as compared to investment in risk-free asset.

  7. Resulting Equilibrium Conditions (cont’d) 3.The risk premium on the market portfolio will be proportional to its risk and the degree of the average risk aversion of all market participants. While M is the efficient diversified optimal portfolio, unsystematic risk is eliminated, market variance denote just the systematic risk.

  8. 4. Risk premium on an individual security is a function of its covariance with the market. The risk premium on individual assets will be proportional to the risk premium on the market portfolio and the beta coefficient of the security on the market port. Beta measures the extent to which returns on the stock and the market move together.

  9. Capital Market Line E(r) CML M E(rM) rf  m

  10. Slope and Market Risk Premium M = Market portfolio rf = Risk free rate E(rM) - rf = Market risk premium E(rM) - rf = Market price of risk = Slope of the CAPM  M

  11. Why all investors would hold the Market Portfolio? • With the these assumptions, investors all must arrive at the same determination of the optimal risky portfolio. They all derive identical efficient frontiers and find the same portfolio for the CAL from T-bills to the frontier.

  12. If the weight of GM stock in each common risky port is 1%, then GM also will comprise 1% of the market portfolio. The same principle applies to the proportion of any stock in each investor’s risky portfolio. As a result, the optimal risky port of all investors is simply a share of the market portfolio. Because each investor has the market portfolio for the optimal risky portfolio the CAL is called the CML

  13. Now suppose that the optimal portfolio of the our investors does not include the stock of say Delta Air Lines when no investor is willing to had Delta stock, the demand is zero, and the stock price will fall. As Delta stock gets cheaper, it begins to look more attractive, while all other stocks look less attractive. Delta will reach a price at which it is desirable to include it in the optimal portfolio and all investors will buy. This price adjustment process guarantees that all stocks will be included in the optimal portfolio. The only issue is the price. If all investors had an identical risk portfolio, this portfolio must be the market portfolio.

  14. The Passive Strategy is Efficient • The passive strategy, using the CML as the optimal CAL, is a powerful alternative to an active strategy. In the simple world of CAPM, all investors use precious resources in security analysis. A passive investor’s investing in the market portfolio benefits from the efficiency of that portfolio. An active investor who chooses any other portfolio will end on a CAL that is less efficient than the CML.

  15. Mutual Fund Theorem • Only one mutual fund of risky assets –the market portfolio- is enough to satisfy the investment demand of all investors. • All investors agree that the mutual fund theorem is that a passive investors may view the market index as a reasonable first approximation to an efficient risky portfolio.

  16. The Risk Premium of the Market Portfolio • The equilibrium risk premium of the market portfolio will be proportional to the degree of average A and σm2. • Each individual investor chooses a proportion y, allocated to the optimal portfolio, M such that;

  17. The Risk Premium of the Market Portfolio • In the simplified CAPM economy, risk free investments involve borrowing and lending among investors. Any borrowing position must be offset by the lending position. This means that net borrowing and lending across all investors must be zero and the average position in the risky portfolio is 100 %, y = 1. Setting y = 1 to equation above;

  18. Expected Returns on Individual Securities • The contribution of one stock to the portfolio variance can be expressed as the sum of all the covariance terms. • For example the contribution of GM stock to the variance of the market portfolio is:

  19. Expected Returns on Individual Securities • We can best measure the stock’s contribution to the risk of the market portfolio by its covariance with the market portfolio GM’s contr to mrk variance = wGMCov(rGM,rm)

  20. Expected Returns on Individual Securities • The reward-to-risk ratio for investments in GM; GM’s contribution to the risk premium/ GM’s contribution to the variance

  21. Expected Returns on Individual Securities • The market port is the tangency (efficient mean-variance) port. The reward-to-risk ratio for investment in the market port. • Market risk premium/market variance (Market price of risk) =

  22. Expected Returns on Individual Securities • A basic principal of equilibrium is that all investments should offer the same reward-to-risk ratio. So that the reward-to-risk ratios of GM and the market port should be equal;

  23. Expected Returns on Individual Securities • This is expected return and beta relationship which is the expression of the CAPM.

  24. Security Market Line • We can view the expected return and beta relationship in a reward-risk equation. The beta of a security is the appropriate measure of its risk because beta is proportional to the risk the security contributes to the optimal risky. • We would expect the reward (risk premium) on individual assets, to depend on the risk an individual asset contributes to the overall portfolio. Because the beta of a stock measures the stock’s contribution to the variance of the market portfolio, we expect the required risk premium to be a function of beta. Security’s risk premium is directly proportional to both the beta and the risk premium of the market portfolio.

  25. Security Market Line E(r) SML E(rM) rf b bM = 1.0

  26. SML Relationships = [COV(ri,rm)] / m2 Slope SML = E(rm) - rf = market risk premium SML = rf + [E(rm) - rf] Betam = [Cov (rm,rm)] / sm2 = sm2 / sm2 = 1

  27. Sample Calculations for SML E(rm) - rf = .08 rf = .03 x = 1.25 E(rx) = .03 + 1.25(.08) = .13 or 13% y = .6 E(ry) = .03 + .6(.08) = .078 or 7.8%

  28. Graph of Sample Calculations E(r) SML Rx=13% .08 Rm=11% Ry=7.8% 3% b .6 by 1.0 1.25 bx

  29. Differences between CML and SML • The CML graphs the risk premiums of efficient portfolios (ports composed of market and risk free asset) as a function of portfolio standard deviation. This is appropriate because standard deviation is a valid measure of risk for efficiently diversified portfolios that are candidates for an investor’s overall portfolio. • The SML graphs the risk premium of individual assets as a function of asset risk. The relevant measure of risk for individual assets is not the asset’s standard deviation instead the contribution of the asset to the portfolios standard deviation as measured by the asset’s beta. The SML is valid both for portfolios and individual assets.

  30. Disequilibrium Example E(r) SML 15% Rm=11% rf=3% b 1.25 1.0

  31. Disequilibrium Example (cont.) • Suppose a security with a  of 1.25 is offering expected return of 15%. (Actually expected return) • According to SML, it should be 13%. (Fair Expected Return) • Under-priced: offering too high of a rate of return for its level of risk. • α (Alpha) = 15%-13%=2% (positive for cheap securities).

  32. Black’s ZeroBeta Model • Absence of a risk-free asset: When investors can not borrow at rf, they may choose risky portfolios from the entire set of efficient frontier portfolios according to how much risk they choose to bear. • The market is no longer the common optimal portfolio.

  33. Black’s Zero Beta Model • An equilibrium expected return-beta relationship in the case of restrictions on risk free has been developed by Fischer Black. • Black’s model of the CAPM in the absence of risk free rests on 3 properties of mean-variance efficient ports. • Combinations of portfolios on the efficient frontier are efficient. • All frontier portfolios have companion portfolios (on the bottom half-the inefficient part) that are uncorrelated. Because the portfolios are uncorrelated, the companion portfolio is referred to as the zero-beta portfolio of the efficient frontier portfolio. • Returns on individual assets can be expressed as linear combinations of efficient portfolios.

  34. Black’s Zero Beta Model Formulation

  35. Efficient Portfolios and Zero Companions E(r) Q P E[rz (Q)] Z(Q) Z(P) E[rz (P)] s

  36. Zero Beta Market Model with Rf Lending but not Borrowing We can express express the return on any security in terms of M and Z (M) where P has been replaced by M, Q has been replaced by Z(M) and rf has been replaced with E(rZ(M)) as; CAPM with E(rz (m)) replacing rf Cov(rM,rZ(M))=0

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